`int_0^3(2sin(x) - e^x)dx` Evaluate the integral.

Textbook Question

Chapter 5, 5.3 - Problem 34 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the definite integral using the fundamental theorem of calculus, such that:

`int_a^b f(u) du = F(b) - F(a)`

`int_0^3 (2sin x - e^x) dx =int_0^3 2sin x dx - int_0^3 e^x dx`

`int_0^3 2sin x dx = 2(-cos x)|_0^3 = -2cos 3 + 2cos 0 = 2 - 2cos 3`

`int_0^3 e^x dx = e^x|_0^3 = e^3 - e^0 = e^3 - 1`

Gathering the results yields:

`int_0^3 (2sin x - e^x) dx = 2 - 2cos 3 - e^3 + 1 = 3 - 2cos 3 - e^3`

Hence, evaluating the definite integral, yields `int_0^3 (2sin x - e^x) dx = 3 - 2cos 3 - e^3.`

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